#7877196

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#51410	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4376 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

(1,2,5,4)(3,7,8,6)(9,10,15,17)(11,20,23,16)(12,18,24,14)(13,21,22,19)(25,26,30,35)(27,31,36,32)(28,37,38,33)(29,34,39,40);(1,3,5,8)(2,6,4,7)(9,11)(10,16)(12,21)(13,18)(14,22)(15,23)(17,20)(19,24)(25,27)(26,31)(28,34)(29,33)(30,36)(32,35)(37,39)(38,40);(1,2,5,4)(3,6,8,7)(10,17)(12,24)(16,20)(19,21)(25,27,30,36)(26,32,35,31)(28,34,38,40)(29,33,39,37);(1,2,5,4)(3,6,8,7)(9,12)(10,18)(11,21)(13,16)(14,17)(15,24)(19,23)(20,22)(25,27)(26,31)(28,34)(29,33)(30,36)(32,35)(37,39)(38,40);(9,13,15,22)(10,19,17,21)(11,18,23,14)(12,16,24,20)(25,28,30,38)(26,33,35,37)(27,34,36,40)(29,32,39,31);(1,4,5,2)(3,7,8,6)(9,14,15,18)(10,12,17,24)(11,22,23,13)(16,21,20,19)(25,29)(26,34)(27,33)(28,31)(30,39)(32,38)(35,40)(36,37)

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=gh,d^2=ghi,e^2=h,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7784392);